Whisk the lemon zest, lemon juice, sugar, salt in to a medium saucepan. Whisk the whole eggs and egg yolks in a small bowl, then whisk into lemon mixture.
Cook the mixture over medium heat. Don't stop whisking (or you while make eggs). Whisk until it's like a pudding. 6-8 minutes. Remove from heat and whisk in the butter a few pieces at a time.
Strain the mixture through a fine-mesh sieve in to a small bowl. Push it through with a rubber spatula. (TASTE TEST) Press plastic wrap directly on surface preventing skin from forming. Chill at least 2 1/2 hours and up to 5 days.